I have read Introduction to real analysis by Manfred Stoll. And I am confused about this question: Construct a sequence $\{f_{n}\}$ of measurable functions on $[0,1]$ such that $\{f_{n}(x)\}$ converges for each $x\in[0,1]$ but that $\{f_{n}\}$ does not converge uniformly on any measurable set $E\subset[0,1]$ with $\lambda([0,1] - E)=0$.
Is it a contradiction with Egorov's Theorem?
No, it doesn't contradict Egorov. Egorov says if $\epsilon > 0$ you can remove a set of measure $< \epsilon$ and get uniform convergence on the rest. Here the removed set would need to have measure $0$.